Jul

4

Recently, Dr Z chastised justly, with: "What happened to numbers on the table?" Mea culpa. So here is some counting:

Russ's post from a couple of weeks ago, on inside and outside days, got me thinking and counting. Russ's stats showed that there were an unusually small number of Mondays that qualified as outside days. I ran a sim that randomly resorted the actual weekday designations among all the actual days (e.g., a Tuesday might be randomly reassigned to be a Friday, etc.), and then counted the number of Monday-inside days, Monday-outside days, Tuesday-inside days, and so on, for each run of the simulation, to 10,000 runs. The sim results confirmed all of Russ's binomial calculations closely, using SPY data from 1 January, 2000, to 22 June, 2011:

weekday-type/total/%tile
Mon-in  68  84.93%
Mon-out 35   0.01%
Tue-in  65  41.85%
Tue-out 80  99.30%
Wed-in  54   2.77%
Wed-out 82  99.65%
Thu-in  63  36.56%
Thu-out 54  10.77%
Fri-in  72  85.36%
Fri-out 55  15.51%

There were only 35 Monday outside days, and this number was at the
0.01%tile of the distribution of 10,000 sim run results (the minimum total for Monday outside days in the 10,000 sim runs was 34). Why? Is Monday less volatile than the other days? Mr. Sogi did some counting that indicated not. What other factors could be involved?

It turns out that SPY tends to go ex-div on Fridays, so could that be the reason for so few Mondays being outside days? A quick check of the 35 Mondays that followed ex-div Fridays shows that two of them were outside days. That's 5.7%, which is consistent with the 6.4% outside days in the total sample of 543 Mondays. So no indication that ex-div Fridays are a factor.

But inside and outside days are like a basketball going through a basket: the ball has to be small enough to fit through the basket, the basket big enough to let the ball through. But what if you also
*move* the basket? To account for "basket movement", we also have to look at the Close-to-Open moves preceding the days in question:

SD of Close-Open % moves:

(Note that holidays are included, so that some "Mon-Tue" moves are actually "Fri-Tue" moves, etc.)

Fri-Mon: 0.829%
Mon-Tue: 0.705%
Tue-Wed: 0.626%
Wed-Thu: 0.633%
Thu-Fri: 0.811%

So we see that the SD of the Friday-Monday move is the largest. To explore the effect of the Close-Open gap, I ran another simulation that used the actual Fridays and Mondays, but replaced the actual Close-Open % move with a value drawn from a normal distribution with the same mean and SD (+0.038% and 0.829%, respectively) as the actual Friday-Monday Close-Open values. The sim randomly assigned new Close-Open % changes to the actual data and then counted the number of inside and outside days in each of 1000 runs:

mean # inside days: 58.4, sd: 6.08
mean # outside days: 40.8, sd: 5.27

Then I ran the sim again, changing the SD of the Close-Open distribution from 0.829%:

SD: 0.629%
mean # inside days: 64.6, sd: 6.15
mean # outside days: 44.7 sd: 5.27

SD: 0.429%
mean # inside days: 71.1, sd: 6.32
mean # outside days: 49.4 sd: 5.21

We can see that the number of inside and outside days increases as the Close-Open gap gets smaller.

When the SD for the Close-Open distribution is set to the actual value of 0.829%, the sim doesn't produce the same result as the actual data. This is because the Close-Open % changes are not normally distributed; rather, the distribution has long tails. Here are probability plots for three sets of Close-Open % moves for the actual SPY data: Friday-Monday, Monday-Tuesday, and Tuesday-Wednesday:

The distribution of Close-Open moves for Friday-Monday has a greater range and denser tails, and the additional extreme values, or extra "basket movement", reduces the number of Monday outside days.

Monday also has the smallest mean High-Low range of all the weekdays:

mean High-Low ranges:

Mon    1.502%
Tue    1.571%
Wed    1.602%
Thu    1.636%
Fri    1.559%

So, between the greater volatility of the Friday-Monday Close-Open move, and the smaller Monday High-Low range, we seem to have a pretty good explanation for the low count of Monday outside days.


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